Observer-based bipartite consensus tracking of nonlinear fractional-order multi-agent systems with pull-based dynamic event-triggered mechanism

Abstract

This paper investigates the observer-based bipartite consensus tracking of nonlinear fractional-order multi-agent systems (FOMASs) by employing a pull-based dynamic event-triggered mechanism (DETM). First, considering that the relevant state information of each agent is not always measurable, a class of distributed observers is considered for each agent to estimate its state information. Then, a pull-based DETM for FOMASs is proposed to avoid continuous controller updates, in which the dynamic threshold is modulated according to the preset conditions. The pull-based DETM constructed in this paper enables each agent to update the controller only based on its own trigger instants. Furthermore, an observer-based dynamic event-triggered control protocol is designed to guarantee the bipartite consensus tracking of FOMASs. Correspondingly, sufficient conditions are obtained by using graph theory and choosing suitable Lyapunov candidate functions. Moreover, the Zeno behavior is precluded. Finally, two simulation examples are presented to illustrate the theoretical results efficiently.

Keywords

Nonlinear fractional-order multi-agent systems pull-based dynamic event-triggered mechanism observer-based bipartite consensus tracking

Introduction

With the development of science and technology, cooperative control of multi-agent systems (MASs) has received extensive attention in the fields of social science, computer science, engineering, and ecology. However, most of the literature focuses on the integer-order dynamics (Huang et al., 2021; Wang et al., 2022; Yan et al., 2022; Zhan et al., 2023; Zhang et al., 2021b). In fact, for dynamics problems such as viscoelastic dampers (Ray et al., 2016), sinusoidal oscillators (Mishra et al., 2018), and seismic transducers (Veeraian et al., 2018), traditional integer-order dynamics can no longer accurately express them. Fractional-order calculus, as a generalization of integer-order calculus, has received increasing attention due to the fact that it effectively describes the long-range dependence of various materials and processes, such as the voltage-current relation of semi-infinite lossy transmission line and heat diffusion over semi-infinite solid. Furthermore, when modeling some natural properties, such as the electrical properties of materials and the rheological properties of rocks, can be better described by fractional-order dynamics. So far, numerous results on fractional-order multi-agent systems (FOMASs) (Bai et al., 2018; Wang and Dong, 2023; Xia et al., 2023; Zhang et al., 2019) have been available.

Generally speaking, most of the research assumes that the information interactions between the agents are cooperative, that is, the edge weights of the communication network graph between the agents are all non-negative. However, in some real-world scenarios, for example, social networks (Easley and Kleinberg, 2010), opinion dynamics (Altafini and Lini, 2014), and biological systems, cooperation and competition exist at the same time, that is, the edge weights of the communication network graph between agents are both positive and negative. The concept about the bipartite consensus was proposed for the first time in Altafini (2012). After that, numerous results have been obtained in the field of bipartite consensus (Lu et al., 2022; Sakthivel et al., 2023; Zhao et al., 2023). For the FOMASs, Gong (2020) discussed the problem of exponential bipartite consensus under switching-directed topology, and Shahvali et al. (2020) investigated the bipartite consensus via utilizing the fully distributed controller. Up to now, to the best of our knowledge, there are few results on bipartite consensus for FOMASs. This is another motivation to promote this article. It should be mentioned that Lu et al. (2022), Gong (2020) required continuous controller updates, which may lead to excessive energy consumption for practical applications.

Since the event-triggered mechanism (ETM) will trigger the control task only when the system states satisfy the given event-triggering conditions, it can effectively save resources, and fruitful conclusions about the ETM have been achieved by Yu et al. (2022), Zhang et al. (2021a), and Jiang et al. (2022). Yu et al. (2022) proposed a mode-dependent ETM to reduce communication and computational load. In order to achieve intermittent communication among neighbors, an innovative adaptive ETM was designed by Zhang et al. (2021a). In addition, the fully distributed controllers by utilizing the pull-based ETM were designed by Jiang et al. (2022) to reduce controller updates. Although the ETM can effectively reduce resources in the beginning, events will be triggered more frequently as the error becomes smaller and smaller. To cope with this matter, the DETM was imported in Girard (2014), which means that the threshold for DETM can adjusted. Furthermore, some corresponding results have been obtained by Chai et al. (2022) and Xu et al. (2023). Known by the authors, up to now, for the FOMASs, there is no paper involving the distributed bipartite consensus tracking driven by pull-based DETM. Therefore, the problem of bipartite consensus tracking with pull-based DETM for FOMASs is still open and awaits a breakthrough, which is also one of the main motivations for promoting this article.

Motivated by the above discussion, an observer-based bipartite consensus tracking problem is considered for FOMASs by employing pull-based DETM. To the best of the authors’ knowledge, there are few papers investigating the problem of FOMASs. The main contributions of this paper can be summarized as follows. First, considering that the relevant state information of each agent is not always measurable, a class of distributed observers is considered for FOMASs. Then, in contrast to continuous controller updates in the existing literature on FOMASs Liu et al. (2023) and Mahmoodi and Shojaei (2022), the pull-based dynamic event-triggered control protocol is designed based on the estimated information, which enables each agent to update the controller only based on its own trigger instants, that is, it can efficiently decrease controller updates. Moreover, the dynamic threshold of the DETM can modulated dynamically according to the preset conditions, which guarantees the larger inter-event intervals.

The remainder of this article can be summarized as follows. The next section presents the elementary preliminaries and problem formulation. Then, the observer-based bipartite consensus tracking problem under the continuous-time control protocol and the pull-based dynamic event-triggered control protocol are investigated, respectively. There follows a section that presents numerical simulation experiments to validate the theoretical results. The final section summarizes the whole article.

Preliminaries and problem formulation

Graph theory

A signed communication graph $\overset{⌣}{G} = (\overset{⌣}{W}, \overset{⌣}{R})$ is considered, where $\overset{⌣}{W} = {1, 2, \dots, N}$ is the set of nodes, $\overset{⌣}{R} \subseteq \overset{⌣}{W} \times \overset{⌣}{W}$ is the set of edges, and matrix $\overset{⌣}{A} = [a_{ij}]_{N \times N}$ is called the adjacency matrix. An edge $(j, i) \in \overset{⌣}{R}$ implies that $a_{ij} \neq 0$ , where $a_{ij} > 0$ and $a_{ij} < 0$ represent the cooperative interaction and antagonistic interaction between agent i and j, respectively. The Laplacian matrix $\overset{⌣}{L}$ can be given as $\overset{⌣}{L} = S - \overset{⌣}{A}$ , where $S = diag (s_{1}, s_{2}, \dots, s_{N})$ , and $s_{i} = \sum_{j = 1}^{N} | a_{ij} |$ . In the leader-following case, consider a diagonal matrix $H = diag (a_{10}, a_{20}, \dots, a_{N 0})$ , where $a_{i 0} = 1$ represents that agent i can receive the information from leader agent, otherwise $a_{i 0} = 0$ , $L_{H}$ is defined as $L_{H} = \overset{⌣}{L} + H$ .

Definition 1. (Altafini, 2012) A signed communication graph $\overset{⌣}{G}$ is referred to as structurally balanced, if $\overset{⌣}{W}$ can be classified into two disjoint subsets ${\overset{⌣}{W}}_{1}, {\overset{⌣}{W}}_{2}$ , satisfying ${\overset{⌣}{W}}_{1} \cup {\overset{⌣}{W}}_{2} = \overset{⌣}{W}$ , ${\overset{⌣}{W}}_{1} \cap {\overset{⌣}{W}}_{2} = \emptyset, a_{ij} \geq 0, i, j \in {\overset{⌣}{W}}_{q}$ and $a_{ij} \leq 0, i \in {\overset{⌣}{W}}_{q}, j \in {\overset{⌣}{W}}_{3 - q}, (q \in {1, 2})$ .

Lemma 1. (Altafini, 2012) A signed communication graph $\overset{⌣}{G}$ is structurally balanced if and only if $\exists \hat{D} \in \hat{D} = diag {(d_{1}, d_{2}, \dots, d_{N})}$ , where $d_{i} \in {1, - 1}$ . Moreover, the bipartition of $\overset{⌣}{W}$ depends on $\hat{D}$ , such as ${\overset{⌣}{W}}_{1} = {i | d_{i} = 1}$ and ${\overset{⌣}{W}}_{2} = {i | d_{i} = - 1}$ .

Assumption 1. The signed communication graph $\overset{⌣}{G}$ is connected and structurally balanced.

Lemma 2. (Zhang et al., 2020) If Assumption 1 holds, then $\exists \hat{D} \in \hat{D}$ such that $H^{*} = \hat{D} L_{H} \hat{D}$ is positive definite and $L^{*} = \hat{D} L \hat{D}$ is semipositive definite.

Caputo fractional derivative and its properties

Definition 2. (Podlubny, 1999) The Caputo fractional derivative of function $f \in C^{m} ([t_{0}, \infty], R)$ with order $α \in (m - 1, m]$ can be given as

D^{α} f (t) = \frac{1}{Γ (m - α)} \int_{0}^{t} (t - ℓ)^{m - α - 1} f^{(m)} (ℓ) d ℓ

(1)

where $Γ (α) = \int_{0}^{\infty} t^{α - 1} e^{- t} dt$ . Note that $D^{α} f (t) = \frac{1}{Γ (1 - α)} \int_{0}^{t} (t - ℓ)^{- α} f' (ℓ) d ℓ$ holds for $0 < α < 1$ .

Definition 3. (Podlubny, 1999) The Caputo fractional integral of function $f \in C^{m} ([t_{0}, \infty], R)$ with order $α \in (0, 1)$ is defined as

I^{α} f (t) = \frac{1}{Γ (α)} \int_{0}^{t} {(t - ℓ)}^{α - 1} f (ℓ) d ℓ

(2)

Lemma 3. (Podlubny, 1999) Consider a continuous function $f (t)$ , if $0 < α < 1$ , then the following equality holds

I^{α} D^{α} f (t) = f (t) - f (0)

(3)

Lemma 4. (Liu et al., 2022) For $\forall t > t_{0} \geq 0$ , if ${\begin{matrix} lim_{t \to \infty} { \end{matrix}}_{t_{0}} I_{t}^{α} {| f (t) |}^{p} \leq M$ , then $lim_{t \to \infty} f (t) = 0$ , where $0 < α < 1$ , $f (t)$ is a continuous function, and p and M are positive constants.

Definition 4. (Wang and Dong, 2023) For $c > 0$ and $d > 0$ , the Mittag-Leffler function can be given as

E_{c, d} (ϑ) = \sum_{k = 0}^{\infty} \frac{ϑ^{k}}{Γ (kc + d)}

(4)

where $ϑ \in C$ and $E_{1, 1} (ϑ) = e^{ϑ}$ . Denote $E_{c, 1} (ϑ) = E_{c} (ϑ)$ .

Lemma 5. (Wang and Dong, 2023) If $c \in (0, 1]$ , $d > 0$ and $t > 0$ , then $0 < E_{c} (- d t^{c}) < 1$ , $E_{c, c} (- d t^{c}) > 0,$ and $0 < t^{c} E_{c, c + 1} (- d t^{c}) < \frac{1}{d}$ .

Lemma 6. (Wang and Dong, 2023) For any constants $α \in (0, 1]$ and $q \in R$ , considering a continuous function $ℏ (t)$ , the solution of $D_{t}^{α} y (t) = qy (t) + ℏ (t)$ is presented as

\begin{matrix} y (t) = y (t_{0}) E_{α} (q {(t - t_{0})}^{α}) \\ + \int_{t_{0}}^{t} (t - ℓ)^{α - 1} E_{α, α} (q {(t - ℓ)}^{α}) ℏ (ℓ) d ℓ \end{matrix}

(5)

Lemma 7. (Wang and Dong, 2023) Consider a differentiable and continuous function $y (t) \in R^{n}$ . For $ℓ > 0$ , $0 < α < 1$ , and matrix $Q > 0$ , the following inequality holds

D^{α} (y^{T} (ℓ) Qy (ℓ)) \leq 2 y^{T} (ℓ) Q D^{α} y (ℓ)

(6)

Lemma 8. (Gong, 2020) For any vectors $w_{1}, w_{2} \in R^{n}$ , the following relationship holds

2 w_{1}^{T} EF w_{2} \leq w_{1}^{T} EG E^{T} w_{1} + w_{2}^{T} F^{T} G^{- 1} F w_{2}

(7)

where matrices $G > 0$ , E and F with appropriate dimensions.

Problem formulation

Consider a nonlinear FOMAS consisting of one leader and N followers, in which one leader is denoted as 0. The dynamic model of a leader is described by

\begin{matrix} D^{α} x_{0} (t) = A x_{0} (t) + f (x_{0} (t)) \\ y_{0} (t) = C x_{0} (t) \end{matrix}

(8)

where $0 < α < 1$ , and $x_{0} (t) \in R^{n}$ and $y_{0} (t) \in R^{p}$ represent the state and output of leader 0, respectively.

The dynamic model of follower i is characterized by

\begin{matrix} D^{α} x_{i} (t) = A x_{i} (t) + B u_{i} (t) + f (x_{i} (t)) \\ y_{i} (t) = C x_{i} (t), i \in F \end{matrix}

(9)

where $0 < α < 1$ , $x_{i} (t) \in R^{n}$ , $u_{i} (t) \in R^{m}$ , and $y_{i} (t) \in R^{p}$ represent the state, control input, and output of follower $i$ , respectively. $F = {1, \dots, N}$ presents the set consisting of N followers.

Remark 1. Compared with the work of Yan et al. (2022), Wang et al. (2022), and Zhan et al. (2023) which concentrate on the bipartite control of integer-order MASs, this paper investigates more general systems, that is, FOMASs. Fractional-order models are more applicable than integer-order models in some practical applications due to its pragmatic significance of capturing non-local behavior and memory retention phenomena.

Assumption 2. The matrix pair $(A, B)$ is stabilizable.

Assumption 3. The matrix pair $(A, C)$ is detectable.

Assumption 4. For any vectors $x_{1}, x_{2} \in R^{n}$ , there exists a constant $ρ > 0$ such that the vector-valued function $f (\cdot)$ satisfies the following inequality

‖ f (x_{1}) - d_{i} f (x_{2}) ‖ \leq ρ ‖ x_{1} - d_{i} x_{2} ‖

(10)

where $d_{i} \in {- 1, 1}$ .

Remark 2. Assumption 1 is a sufficient condition to achieve bipartite consensus. Assumptions 2 and 3 are basic assumptions for achieving consensus, and they are also demanded by Rong et al. (2021)and Xu et al. (2022). The Lipschitz condition assumption is very important to deal with nonlinear systems. Many functions satisfy the above Lipschitz condition (10), which is also used by Li et al. (2021)and Gong (2020).

Based on Assumptions 2 and 3, the state observer is given as

\begin{matrix} D^{α} ϖ_{i} (t) = A ϖ_{i} (t) + B u_{i} (t) + f (ϖ_{i} (t)) + R (C ϖ_{i} (t) - y_{i} (t)) \\ D^{α} ϖ_{0} (t) = A ϖ_{0} (t) + f (ϖ_{0} (t)) + R (C ϖ_{0} (t) - y_{0} (t)) \end{matrix}

(11)

where $ϖ_{i} (t)$ is the estimation of $x_{i} (t)$ .

Definition 5. The FOMASs reach bipartite consensus tracking, if for any initial conditions,

lim_{t \to \infty} ∥ x_{i} (t) - d_{i} x_{0} (t) ∥ = 0, i \in F

(12)

where $d_{i} = 1$ or $- 1$ .

Observer-based bipartite consensus tracking via continuous-time control protocol

In this section, a distributed observer-based continuous-time controller will be designed to arrive at the bipartite consensus tracking of nonlinear FOMAS (9) with the leader (8).

Denote the combined measurement as

g_{i} (t) = \sum_{j = 1}^{N} | a_{ij} | (ϖ_{i} (t) - sgn (a_{ij}) ϖ_{j} (t)) + a_{i 0} (ϖ_{i} (t) - d_{i} ϖ_{0} (t))

(13)

where $sgn (\cdot)$ represents the sign function. The observer-based distributed continuous-time controller is defined as follows

u_{i} (t) = K g_{i} (t)

(14)

where $K \in R^{m \times n}$ is the feedback gain matrix.

Remark 3. It is well known that the state information of agent is hard to measure in practice applications. To avoid this problem, this article presents a fractional-order state observer (11) to estimate the state information of each agent. Moreover, the controller based on the observation information is designed; in this way, it is more reasonable in practical applications.

Theorem 1. Suppose that Assumptions 2 and 3 are satisfied, there exists a symmetric positive definite matrix $P$ satisfying

P A + A^{T} P - 2 \underline{λ} P B B^{T} P + σ P P + Ω I_{n} = 0

(15)

where $K = - B^{T} P$ , $Ω = 1 + \frac{ρ_{1}}{σ} + ϵ {\bar{λ}}^{3}$ , $\bar{λ} = λ_{max} (L_{H}) > 0$ , and $σ$ , $ϵ$ , $ρ_{1}$ , and $ρ_{2}$ are positive constants. $R = - Q_{1} C^{T}$ is designed such that $A + RC + ρ_{2} I_{n}$ is a fractional-order Hurwitz matrix, where $Q_{1} > 0$ is the solution of the equality $Q_{1} A^{T} + A Q_{1} - Q_{1} C C^{T} Q_{1} + I_{n} = 0$ . The bipartite consensus tracking of nonlinear FOMAS (9) with the leader (8) can be arrived under the observer (11) and the observer-based control law (14).

Proof. Define $℘_{i} (t) = x_{i} (t) - d_{i} x_{0} (t), i \in F$ , $℘ (t) = [℘_{1}^{T} (t), ℘_{2}^{T} (t), \dots, ℘_{N}^{T} (t)]^{T}$ . Define ${\hat{℘}}_{i} (t) = ϖ_{i} (t) - d_{i} ϖ_{0} (t), i \in F$ , $\hat{℘} (t) = [{\hat{℘}}_{1}^{T} (t), {\hat{℘}}_{2}^{T} (t), \dots, {\hat{℘}}_{N}^{T} (t)]^{T}$ . Denote the measurement error ${\tilde{℘}}_{i} (t) = ℘_{i} (t) - {\hat{℘}}_{i} (t), i \in F$ , $\tilde{℘} (t) = [{\tilde{℘}}_{1}^{T} (t), {\tilde{℘}}_{2}^{T} (t), \dots, {\tilde{℘}}_{N}^{T} (t)]^{T}$ . By the definition of $℘ (t)$ , we derive

D^{α} ℘ (t) = (I_{N} \otimes A) ℘ (t) + (L_{H} \otimes BK) (℘ (t) - \tilde{℘} (t)) + ℵ (t)

(16)

where $ℵ (t) = [{(f (x_{1} (t)) - d_{1} f (x_{0} (t)))}^{T}, {(f (x_{2} (t)) - d_{2} f (x_{0} (t)))}^{T}$ $, \dots, {(f (x_{N} (t)) - d_{N} f (x_{0} (t)))}^{T}]^{T}$ .

Let $V_{11} (t) = ℘^{T} (t) (L_{H} \otimes P) ℘ (t)$ , and $V_{12} (t) = \frac{{\bar{μ}}^{2}}{ϵ} {\tilde{℘}}^{T} (t) (I_{N} \otimes Q_{2}) \tilde{℘} (t)$ , where $ϵ$ , $\bar{μ} = λ_{max} (P B B^{T} P) > 0$ , and matrix $Q_{2} > 0$ satisfies

Q_{2} (A + RC + ρ_{2} I_{n}) + {(A + RC + ρ_{2} I_{n})}^{T} Q_{2} + I_{n} = 0

(17)

Consider the Lyapunov function

V_{1} (t) = V_{11} (t) + V_{12} (t)

(18)

Evidently, $V_{1} (t)$ is positive definite. Using the result in Lemma 7, we derive

\begin{matrix} D^{α} V_{11} (t) \leq 2 ℘^{T} (t) (L_{H} \otimes P) D^{α} ℘ (t) \\ \leq ℘^{T} (t) (L_{H} \otimes (P A + A^{T} P) - 2 L_{H}^{2} \otimes P B B^{T} P) ℘ (t) \\ + 2 ℘^{T} (t) (L_{H}^{2} \otimes P B B^{T} P) \tilde{℘} (t) + 2 ℘^{T} (t) (L_{H} \otimes P) ℵ (t) \end{matrix}

(19)

By employing Lemma 8, it derives that

\begin{array}{l} 2 ℘^{T} (t) (ℒ_{ℋ}^{2} \otimes P B B^{T} P) \tilde{℘} (t) \leq ϵ ℘^{T} (t) (ℒ_{ℋ}^{4} \otimes I_{n}) ℘ (t) \\ + \frac{1}{ϵ} {\tilde{℘}}^{T} (t) (I_{N} \otimes {(P B B^{T} P)}^{2}) \tilde{℘} (t) \\ \leq ϵ {\bar{λ}}^{3} ℘^{T} (t) (ℒ_{ℋ} \otimes I_{n}) ℘ (t) + \frac{{\bar{μ}}^{2}}{ϵ} {\tilde{℘}}^{T} (t) \tilde{℘} (t) \end{array}

(20)

and

\begin{matrix} 2 ℘^{T} (t) (L_{H} \otimes P) ℵ (t) \leq σ ℘^{T} (t) (L_{H} \otimes P P) ℘ (t) + \frac{1}{σ} ℵ^{T} (t) \\ (L_{H} \otimes I_{n}) ℵ (t) \end{matrix}

(21)

where $ϵ$ and $σ$ are positive constants.

By recalling Lemma 7, Assumption 4, the definition of $\tilde{℘} (t)$ , and equation (17), it is easy to obtain that

\begin{matrix} D^{α} V_{12} (t) \leq \frac{2 {\bar{μ}}^{2}}{ϵ} \sum_{i = 1}^{N} {\tilde{℘}}_{i}^{T} (t) Q_{2} ((A + RC) {\tilde{℘}}_{i} (t) + f (x_{i} (t)) - d_{i} f (x_{0} (t)) \\ - f (ϖ_{i} (t)) + d_{i} f (ϖ_{0} (t))) \\ \leq \frac{{\bar{μ}}^{2}}{ϵ} {\tilde{℘}}^{T} (t) \\ (I_{N} \otimes ((A + RC + ρ_{2} I_{n})^{T} Q_{2} + Q_{2} (A + RC + ρ_{2} I_{n}))) \tilde{℘} (t) \\ = - \frac{{\bar{μ}}^{2}}{ϵ} {\tilde{℘}}^{T} (t) \tilde{℘} (t) \end{matrix}

(22)

Choose an orthogonal matrix $U \in R^{N \times N}$ ensuring $U^{T} L_{H} U = ϒ = diag {λ_{1} (L_{H}), \dots, λ_{N} (L_{H})}$ , where $λ_{i} (L_{H})$ are eigenvalues of $L_{H}$ . Let $\bar{℘} (t) = [{\bar{℘}}_{1}^{T} (t), \dots, {\bar{℘}}_{N}^{T} (t)]^{T} = (U^{T} \otimes I_{n}) ℘ (t)$ , we have

\begin{matrix} D^{α} V_{1} (t) = D^{α} V_{11} (t) + D^{α} V_{12} (t) \\ \leq \sum_{i = 1}^{N} λ_{max} (L_{H}) {\bar{℘}}_{i}^{T} (t) \\ (P A + A^{T} P - 2 \underline{λ} P B B^{T} P + σ P P + (ϵ {\bar{λ}}^{3} + \frac{ρ_{1}}{σ}) I_{n}) {\bar{℘}}_{i} (t) \\ \leq - \bar{λ} ‖ ℘ (t) ‖^{2} \end{matrix}

(23)

which indicates that $V_{1} (t)$ is bounded. According to equation (23), Definition 3, and Lemma 3, one has

\frac{1}{Γ (α)} \int_{0}^{t} (t - ℓ)^{α - 1} ‖ ℘ (ℓ) ‖^{2} d ℓ \leq \frac{1}{\bar{λ}} (V_{1} (0) - V_{1} (t))

(24)

In light of Lemma 4, we can observe that system (16) is asymptotically stable, that is to say, $lim_{t \to \infty} ℘ (t) = 0$ . It means that the nonlinear FOMAS (9) with the leader (8) can arrive bipartite consensus tracking by employing the continuous-time control law. This proof is completed.

Observer-based bipartite consensus tracking via pull-based dynamic event-triggered control protocol

In the section, a distributed observer-based controller by utilizing pull-based DETM will be proposed to realize the bipartite consensus tracking of nonlinear FOMAS (9) with the leader (8), and the Zeno behavior will be precluded.

Denote the measurement error as

ς_{i} (t) = g_{i} (t_{k}^{i}) - g_{i} (t), t \in [t_{k}^{i}, t_{k + 1}^{i})

(25)

where $t_{k}^{i}$ denotes the kth triggering time of the ith agent. Let $ς (t) = [ς_{1}^{T} (t), ς_{2}^{T} (t), \dots, ς_{N}^{T} (t)]^{T}$ . Then, for the nonlinear FOMAS (9), the distributed pull-based dynamic event-triggered controller is defined as follows

u_{i} (t) = K g_{i} (t_{k}^{i})

(26)

where $K \in R^{m \times n}$ is the feedback gain matrix.

Remark 4. In contrast to the work of Zhang et al. (2021a) and Xu et al. (2022) where controllers use the push-based ETM with $\sum_{j = 1}^{N} | a_{ij} | (x_{i} (t_{k}^{i}) - x_{j} (t_{k'}^{j}))$ , where $k' = \arg min_{k \in N : t ⩾ t_{l}^{j}} {t - t_{l}^{j}}$ . This article proposes the controller (26) by employing the pull-based DETM, which means that each agent only updates the controller based on its own trigger instants. Therefore, compared with push-based ETM, this paper uses the pull-based DETM can greatly decrease the frequency of controller updates.

The DETM is proposed as follows

\begin{matrix} t_{0}^{i} = 0 \\ t_{k + 1}^{i} = inf {t > t_{k}^{i} | J_{i} (t) > 0} \end{matrix}

(27)

where $J_{i} (t) = δ_{i} (‖ ς_{i} (t) ‖^{2} - κ_{i} ‖ g_{i} (t) ‖^{2}) - ϕ_{i} (t)$ , $κ_{i} = \frac{ϱ (1 - ν_{i})}{{\bar{μ}}^{2} \bar{λ} (1 + τ)}$ with $0 < ν_{i} < 1$ for $i \in F$ , $\bar{μ} = λ_{max} (P B B^{T} P) > 0$ , $δ_{i} > 0$ , and $ϱ$ and $τ$ are positive constants. The dynamics of $ϕ_{i} (t)$ is characterized by

D^{α} ϕ_{i} (t) = - η_{i} ϕ_{i} (t) + ξ_{i} (κ_{i} {‖ g_{i} (t) ‖}^{2} - {‖ ς_{i} (t) ‖}^{2})

(28)

where $η_{i} > 0$ , $ξ_{i} > 0$ , and $ϕ_{i} (0) > 0$

Lemma 9. For pre-defined parameters $δ_{i}$ , $η_{i}$ , $ξ_{i}$ , and $ϕ_{i} (0) > 0$ , the dynamic variable $ϕ_{i} (t)$ in event-triggered function satisfies

ϕ_{i} (t) > 0, i \in F

(29)

Proof. For $t \in [t_{k}^{i}, t_{k + 1}^{i})$ , by inequality (27), we obtain that

‖ ς_{i} (t) ‖^{2} - κ_{i} ‖ g_{i} (t) ‖^{2} \leq \frac{ϕ_{i} (t)}{δ_{i}}

(30)

It follows from equation (28) that

D^{α} ϕ_{i} (t) \geq - (η_{i} + \frac{ξ_{i}}{δ_{i}}) ϕ_{i} (t)

(31)

Together with Lemmas 5 and 6, one has $ϕ_{i} (t) \geq ϕ_{i} (t_{k}^{i}) E_{α} (- (η_{i} + \frac{ξ_{i}}{δ_{i}}) (t - t_{k}^{i})^{α})$ . Then, when $t \in [t_{k - 1}^{i}, t_{k}^{i})$ , the following inequality holds: $ϕ_{i} (t_{k}^{i}) \geq ϕ_{i} (t_{k - 1}^{i}) E_{α} (- (η_{i} + \frac{ξ_{i}}{δ_{i}}) {(t_{k}^{i} - t_{k - 1}^{i})}^{α})$ . Thus, for $t \in [t_{k - 1}^{i}, t_{k + 1}^{i})$ , one has $ϕ_{i} (t) \geq ϕ_{i} (t_{k - 1}^{i}) E_{α} (- (η_{i} + \frac{ξ_{i}}{δ_{i}}) (t - t_{k - 1}^{i})^{α})$ . Then, for $t \in [0, \infty) = [t_{0}^{i}, t_{1}^{i}) \cup [t_{1}^{i}, t_{2}^{i}) \cup \dots \cup [t_{k}^{i}, t_{k + 1}^{i})$ , it can be further obtained

ϕ_{i} (t) \geq ϕ_{i} (0) E_{α} (- (η_{i} + \frac{ξ_{i}}{δ_{i}}) t^{α}) > 0

(32)

Remark 5. Compared with the work of Mahmoodi and Shojaei (2022) and Xia et al. (2023), one advantage of this article is that a dynamic variable $ϕ_{i} (t)$ is proposed, which is an important factor for adjusting triggering thresholds dynamically. By introducing dynamic variable $ϕ_{i} (t)$ , the problem of events being triggered more frequently in ETM as the error decreases is solved. It means that the number of event-triggered instants is decreased in DETM. Actually, ETM is a special case of DETM, the DETM becomes ETM when $ϕ_{i} (t) = 0$ , $t \in [0, \infty)$ . Therefore, DETM can efficiently decrease the number of event-triggered instants than ETM. In addition, by appropriately increasing the values of $ϱ$ and $ν_{i}$ and decreasing the value of $τ$ , the larger interevent time intervals can be obtained, which also means that the number of event-triggered instants is reduced.

Remark 6. For the existing literature on fractional-order systems, none of Jin et al. (2023) and Xia et al. (2023) contemplated both pull-based event-triggered strategy and DETM. Therefore, we introduce a pull-based DETM in this article, which not only adjusts the dynamic threshold dynamically according to the preset conditions but also updates the controller based on its own trigger instants.

Theorem 2. Suppose that Assumptions 2 and 3 are satisfied, there exists a symmetric positive definite matrix $P$ satisfying

P A + A^{T} P - 2 \underline{λ} P B B^{T} P + σ P P + Π I_{n} = 0

(33)

where $K = - B^{T} P$ , $Π = 1 + \frac{ρ_{1}}{σ} + ϱ \bar{λ} + ϵ {\bar{λ}}^{3}$ , $\bar{λ} = λ_{max} (L_{H}) > 0$ , and $σ$ , $ϵ$ , $ρ_{1}$ , and $ρ_{2}$ are positive constants. $R = - Q_{1} C^{T}$ is designed such that $A + RC + ρ_{2} I_{n}$ is a fractional-order Hurwitz matrix, where $Q_{1} > 0$ is the solution of the equality $Q_{1} A^{T} + A Q_{1} - Q_{1} C C^{T} Q_{1} + I_{n} = 0$ . The bipartite consensus tracking of nonlinear FOMAS (9) with the leader (8) can be arrived under the state observer (11) and the observer-based control protocol (26) driven by DETM (27), if the following inequality holds

η_{i} > \frac{{\bar{μ}}^{2} - ϱ ξ_{i}}{ϱ δ_{i}}

(34)

Proof. Let $V_{21} (t) = ℘^{T} (t) (L_{H} \otimes P) ℘ (t)$ , $V_{22} (t) = Θ {\tilde{℘}}^{T} (t) (I_{N} \otimes Q_{2}) \tilde{℘} (t)$ , and $V_{23} (t) = \sum_{i = 1}^{N} ϕ_{i} (t)$ , where matrix $P$ is defined in Theorem 2, $Θ = \frac{{\bar{μ}}^{2}}{ϵ} + \frac{(1 - ν) \bar{λ}}{τ}$ , and matrix $Q_{2} > 0$ satisfies equation (17).

Consider the Lyapunov function

V_{2} (t) = V_{21} (t) + V_{22} (t) + V_{23} (t)

(35)

Evidently, $V_{2} (t)$ is positive definite according to Lemma 9. By the definition of $℘ (t)$ , we derive

\begin{matrix} D^{α} ℘ (t) = (I_{N} \otimes A) ℘ (t) + (L_{H} \otimes BK) (℘ (t) - \tilde{℘} (t)) \\ + (I_{N} \otimes BK) ς (t) + ℵ (t) \end{matrix}

(36)

According to the result in Lemma 7 and equation (36), we have

\begin{matrix} D^{α} V_{21} (t) \leq 2 ℘^{T} (t) (L_{H} \otimes P) D^{α} ℘ (t) \\ \leq ℘^{T} (t) (L_{H} \otimes (P A + A^{T} P) - 2 L_{H}^{2} \otimes P B B^{T} P) ℘ (t) \\ + 2 ℘^{T} (t) (L_{H}^{2} \otimes P B B^{T} P) \tilde{℘} (t) \\ - 2 ℘^{T} (t) (L_{H} \otimes P B B^{T} P) ς (t) + 2 ℘^{T} (t) (L_{H} \otimes P) ℵ (t) \end{matrix}

(37)

By utilizing Lemma 8, we have

\begin{matrix} - 2 ℘^{T} (t) (L_{H} \otimes P B B^{T} P) ς (t) \leq ϱ ℘^{T} (t) (L_{H}^{2} \otimes I_{n}) ℘ (t) \\ + \frac{1}{ϱ} ς^{T} (t) (I_{N} \otimes {(P B B^{T} P)}^{2}) ς (t) \\ \leq ϱ \bar{λ} ℘^{T} (t) (L_{H} \otimes I_{n}) ℘ (t) + \frac{{\bar{μ}}^{2}}{ϱ} ς^{T} (t) ς (t) \end{matrix}

(38)

where $ϱ$ is a positive constant.

\begin{matrix} D^{α} V_{21} (t) \leq \sum_{i = 1}^{N} λ_{max} (L_{H}) {\bar{℘}}_{i}^{T} (t) \\ (P A + A^{T} P - 2 \underline{λ} P B B^{T} P + σ P P + (ϵ {\bar{λ}}^{3} + \frac{ρ_{1}}{σ} + ϱ \bar{λ}) I_{n}) {\bar{℘}}_{i} (t) \\ + \frac{{\bar{μ}}^{2}}{ϵ} {\tilde{℘}}^{T} (t) \tilde{℘} (t) + \frac{{\bar{μ}}^{2}}{ϱ} ς^{T} (t) ς (t) \\ \leq - \bar{λ} ℘^{T} (t) ℘ (t) + \frac{{\bar{μ}}^{2}}{ϵ} {\tilde{℘}}^{T} (t) \tilde{℘} (t) + \frac{{\bar{μ}}^{2}}{ϱ} ς^{T} (t) ς (t) \end{matrix}

(39)

Taking the $α$ -order derivative of $V_{22} (t)$ gives

D^{α} V_{22} (t) ⩽ - (\frac{{\bar{μ}}^{2}}{ϵ} + \frac{(1 - ν) \bar{λ}}{τ}) {\tilde{℘}}^{T} (t) \tilde{℘} (t)

(40)

Taking the $α$ -order derivative of $V_{23} (t)$ gives

D^{α} V_{23} (t) = \sum_{i = 1}^{N} (- η_{i} ϕ_{i} (t) + ξ_{i} (κ_{i} {‖ g_{i} (t) ‖}^{2} - {‖ ς_{i} (t) ‖}^{2}))

(41)

Combining equations (39)–(41) yields

\begin{matrix} D^{α} V_{2} (t) = D^{α} V_{21} (t) + D^{α} V_{22} (t) + D^{α} V_{23} (t) \\ \leq - \bar{λ} ‖ ℘ (t) ‖^{2} + \frac{{\bar{μ}}^{2}}{ϱ} ς^{T} (t) ς (t) - \frac{(1 - ν) \bar{λ}}{τ} {\tilde{℘}}^{T} (t) \tilde{℘} (t) \\ + \sum_{i = 1}^{N} (- η_{i} ϕ_{i} (t) + ξ_{i} (κ_{i} {‖ g_{i} (t) ‖}^{2} - {‖ ς_{i} (t) ‖}^{2})) \end{matrix}

(42)

where $ν = min {ν_{i}}$ . In view of Lemma 8 and inequality (27), inequality (42) can be further rewritten as

\begin{matrix} D^{α} V_{2} (t) \leq - \bar{λ} ‖ ℘ (t) ‖^{2} + \frac{{\bar{μ}}^{2}}{ϱ} \sum_{i = 1}^{N} ‖ ς_{i} (t) ‖^{2} - \frac{(1 - ν) \bar{λ}}{τ} {\tilde{℘}}^{T} (t) \tilde{℘} (t) \\ + \sum_{i = 1}^{N} (- η_{i} ϕ_{i} (t) + ξ_{i} (κ_{i} {‖ g_{i} (t) ‖}^{2} - {‖ ς_{i} (t) ‖}^{2})) \\ \leq - \bar{λ} ‖ ℘ (t) ‖^{2} + \frac{(1 - ν)}{\bar{λ} (1 + τ)} \sum_{i = 1}^{N} ‖ g_{i} (t) ‖^{2} - \frac{(1 - ν) \bar{λ}}{τ} {\tilde{℘}}^{T} (t) \tilde{℘} (t) \\ + \sum_{i = 1}^{N} (\frac{{\bar{μ}}^{2} - ϱ ξ_{i}}{ϱ δ_{i}} - η_{i}) ϕ_{i} (t) \end{matrix}

\begin{matrix} \leq - \bar{λ} ‖ ℘ (t) ‖^{2} + \frac{(1 - ν)}{\bar{λ}} ℘^{T} (t) (L_{H}^{2} \otimes I_{n}) ℘ (t) \\ + \sum_{i = 1}^{N} (\frac{{\bar{μ}}^{2} - ϱ ξ_{i}}{ϱ δ_{i}} - η_{i}) ϕ_{i} (t) \\ \leq - ν \bar{λ} ‖ ℘ (t) ‖^{2} \end{matrix}

(43)

We can derive that $V (t)$ is bounded. By integrating inequality (43), one can obtain

\frac{1}{Γ (α)} \int_{0}^{t} (t - ℓ)^{α - 1} ‖ ℘ (ℓ) ‖^{2} d ℓ \leq \frac{1}{ν \bar{λ}} (V_{2} (0) - V_{2} (t))

(44)

In light of Lemma 4, we obtain $lim_{t \to \infty} ℘ (t) = 0$ , that is, the observer-based bipartite tracking control problem of FOMASs with pull-based DETM is solved.

We now show that agents are free from Zeno behavior under the proposed pull-based DETM.

Theorem 3. Under the pull-based DETM in Theorem 2, the Zeno behavior can be precluded.

Proof. According to Theorem 2, we can derive that $D^{α} ℘_{i} (t)$ and $D^{α} {\tilde{℘}}_{i} (t)$ are bounded. Based on the Caupto fractional integral and equation (25), we have

\begin{matrix} ‖ ς_{i} (t) ‖ = ‖ g_{i} (t) - g_{i} (t_{k}^{i}) ‖ \\ = ‖ I^{α} (D^{α} g_{i} (t)) - I^{α} (D^{α} g_{i} (t_{k}^{i})) ‖ \\ = ‖ \frac{1}{Γ (α)} (\int_{0}^{t} \frac{D^{α} g_{i} (ℓ)}{{(t - ℓ)}^{1 - α}} d ℓ - \int_{0}^{t_{k}^{i}} \frac{D^{α} g_{i} (ℓ)}{{(t_{k}^{i} - ℓ)}^{1 - α}} d ℓ) ‖ \\ \leq \frac{1}{Γ (α)} ‖ \int_{0}^{t_{k}^{i}} (\frac{D^{α} g_{i} (ℓ)}{{(t - ℓ)}^{1 - α}} - \frac{D^{α} g_{i} (ℓ)}{{(t_{k}^{i} - ℓ)}^{1 - α}}) d ℓ ‖ \\ + \frac{1}{Γ (α)} ‖ \int_{t_{k}^{i}}^{t} \frac{D^{α} g_{i} (ℓ)}{{(t - ℓ)}^{1 - α}} d ℓ ‖ \\ \leq \frac{\bar{λ}}{Γ (α)} ‖ \int_{0}^{t_{k}^{i}} (\frac{D^{α} ℘_{i} (ℓ)}{{(t - ℓ)}^{1 - α}} - \frac{D^{α} ℘_{i} (ℓ)}{{(t_{k}^{i} - ℓ)}^{1 - α}}) d ℓ ‖ \\ + \frac{\bar{λ}}{Γ (α)} ‖ \int_{t_{k}^{i}}^{t} \frac{D^{α} ℘_{i} (ℓ)}{{(t - ℓ)}^{1 - α}} d ℓ ‖ \\ + \frac{\bar{λ}}{Γ (α)} ‖ \int_{0}^{t_{k}^{i}} (\frac{D^{α} {\tilde{℘}}_{i} (ℓ)}{{(t - ℓ)}^{1 - α}} - \frac{D^{α} {\tilde{℘}}_{i} (ℓ)}{{(t_{k}^{i} - ℓ)}^{1 - α}}) d ℓ ‖ \\ + \frac{\bar{λ}}{Γ (α)} ‖ \int_{t_{k}^{i}}^{t} \frac{D^{α} {\tilde{℘}}_{i} (ℓ)}{{(t - ℓ)}^{1 - α}} d ℓ ‖ \end{matrix}

(45)

Since $D^{α} ℘_{i} (t)$ and $D^{α} {\tilde{℘}}_{i} (t)$ are bounded, for any $t \in [t_{k}^{i}, t_{k + 1}^{i})$ , there exists positive constants $M_{1}$ and $M_{2}$ such that $‖ D^{α} ℘_{i} (t) ‖ \leq M_{1}$ , $‖ D^{α} {\tilde{℘}}_{i} (t) ‖ \leq M_{2}$ . Then

\begin{matrix} ‖ ς_{i} (t) ‖ \leq \frac{2 \bar{λ} M}{Γ (α)} | \int_{0}^{t_{k}^{i}} (\frac{1}{{(t - ℓ)}^{1 - α}} - \frac{1}{{(t_{k}^{i} - ℓ)}^{1 - α}}) d ℓ | \\ + \frac{2 \bar{λ} M}{Γ (α)} | \int_{t_{k}^{i}}^{t} \frac{1}{{(t - ℓ)}^{1 - α}} d ℓ | \\ = \frac{2 \bar{λ} M}{Γ (α + 1)} (| {(t - t_{k}^{i})}^{α} - t^{α} + {(t_{k}^{i})}^{α} | + {(t - t_{k}^{i})}^{α}) \\ \leq \frac{4 \bar{λ} M {(t - t_{k}^{i})}^{α}}{Γ (α + 1)} \end{matrix}

(46)

where $M = max {M_{1}, M_{2}}$ .

According to the triggering condition (27), we obtain the next event will not reach before $δ_{i} (‖ ς_{i} (t^{*}) ‖^{2} - κ_{i} ‖ g_{i} (t^{*}) ‖^{2}) = ϕ_{i} (t^{*})$ . Denoting $Λ = t^{*} - t_{k}^{i}$ , one has

\frac{ϕ_{i} (Λ + t_{k}^{i})}{δ_{i}} + κ_{i} ‖ g_{i} (Λ + t_{k}^{i}) ‖^{2} = ‖ ς_{i} (Λ + t_{k}^{i}) ‖^{2} \leq \frac{16 {\bar{λ}}^{2} M^{2} Λ^{2 α}}{Γ^{2} (α + 1)}

(47)

It is obvious that

\begin{matrix} t_{k + 1}^{i} - t_{k}^{i} \geq Λ \geq \\ {[\frac{Γ^{2} (α + 1)}{16 {\bar{λ}}^{2} M^{2}} (\frac{ϕ_{i} (Λ + t_{k}^{i})}{δ_{i}} + κ_{i} {‖ g_{i} (Λ + t_{k}^{i}) ‖}^{2})]}^{\frac{1}{2 α}} > 0 \end{matrix}

(48)

which means that there does not exist infinite event-triggered instants in a finite time. Therefore, Zeno behavior is precluded. Following this, Theorem 3 is completely proved.

Simulation

In this section, two simulation examples are used to certify the validity of the above theoretical results. Consider a FOMAS consisting of one leader and four followers (labeled as 0, 1, 2, 3, and 4). The communication graph among agents is applied as Figure 1. The system matrices of FOMAS (9) with the leader (8) are presented as

\begin{matrix} A = [\begin{matrix} 0 & - 2 \\ 2 & 0 \end{matrix}], B = [\begin{matrix} 1 \\ 1 \end{matrix}], C = [\begin{matrix} 1 \\ 0 \end{matrix}], R = [\begin{matrix} - 0.8592 \\ 0.1095 \end{matrix}] \end{matrix}

Suppose the nonlinear functions are $f (x_{i} (t)) = \frac{\sin (x_{i} (t))}{2}$ , $i = 0, 1, \dots, 4$ . According to Figure 1, the Laplacian matrix among agents is expressed as

\overset{⌣}{L} = [\begin{matrix} 2 & 1 & - 1 & 0 \\ 1 & 3 & 2 & 0 \\ - 1 & 2 & 4 & 1 \\ 0 & 0 & 1 & 1 \end{matrix}]

Example 1. (Continuous-Time Control Law) In this example, we consider the nonlinear FOMAS (9) with the leader (8) under the continuous-time control law. Solving equation (15) with $\underline{λ} = λ_{min} (L_{H}) = 0.3968$ , $\bar{λ} = λ_{max} (L_{H}) = 6.4120$ , $ρ_{1} = 1$ , $σ = 0.2$ , and $ϵ = 0.01$ , we obtain

P = [\begin{matrix} 4.2043 & - 3.6452 \\ - 3.6452 & 11.9144 \end{matrix}]

Moreover, the matrix $K$ is given as

K = [\begin{matrix} - 0.5591 & - 8.2691 \end{matrix}]

The trajectories of state-tracking without control input are applied as Figure 2. Figures 3 and 4 describe the agent trajectories and the observer trajectories, respectively. From Figures 2 and 3, it can be viewed that the design of the controller is effective. Figure 5 depicts the tracking control errors of four agents, we can see from Figure 5 that the bipartite tracking control can be achieved.

Example 2. (Pull-based Dynamic Event-Triggered Control Law) In this example, we consider the nonlinear FOMAS (9) with the leader (8) under the pull-based dynamic event-triggered control law.

Solving equation (33) with $\underline{λ} = λ_{min} (L_{H}) = 0.3968$ , $\bar{λ} = λ_{max} (L_{H}) = 6.4120$ , $ρ_{1} = 1$ , $ϱ = 0.1$ , $σ = 10$ , and $ϵ = 0.001$ , we obtain

P = [\begin{matrix} 1.5309 & - 0.3456 \\ - 0.3456 & 2.8705 \end{matrix}]

Moreover, the matrix $K$ is given as

K = [\begin{matrix} - 1.1853 & - 2.5249 \end{matrix}]

Assume $δ_{1} = δ_{2} = δ_{3} = δ_{4} = 1000$ , $η_{i} = 2.3$ , $ξ_{i} = 0.12$ , and $κ_{i} = 0.0001$ . Suppose the initial values of dynamic parameters are $ϕ_{1} (0) = 3$ , $ϕ_{2} (0) = 5$ , $ϕ_{3} (0) = 2$ , and $ϕ_{4} (0) = 4$ . Figures 6 and 7 show the agent trajectories and the observer trajectories, respectively. The trajectories of bipartite tracking control error are applied as Figure 8. From Figures 6 –8, it can be viewed that all the followers can follow the tracks of the leader with a certain error. Figure 9 gives the evolution of dynamic variable $ϕ_{i} (t)$ , which verifies the effectiveness of Lemma 9. Figure 10 depicts the triggering time for four agents, which illustrates that the Zeno behavior can be precluded. Table 1 exhibits the event-triggered instants within the 20s for the controller (26) and the controller in the work of Xia et al. (2023). It shows that DETM can decrease the number of event-triggered instants than ETM. Therefore, the investigation of DETM is more meaningful than ETM. To conclude, the simulation figures demonstrate the theoretical results efficiently.

Table 1.

Comparison with the static situation.

Agent i	1	2	3	4	Total
ETM (Xia et al., 2023)	694	668	674	655	2691
DETM	170	133	232	175	710

Conclusion

This paper has investigated the observer-based bipartite tracking control of FOMASs by utilizing pull-based DETM. First, an observer has been designed to estimate the state information of each agent. Then, a pull-based DETM has been considered in this paper to avoid continuous controller updates. Furthermore, the distributed continuous-time control protocol and the pull-based dynamic event-triggered control protocol have been proposed for solving the bipartite consensus tracking, respectively. Finally, Zeno behavior has been precluded. As for our future work, it is worth solving the fully distributed DETM bipartite consensus problems of FOMASs.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Fundamental Research Funds for the University of Science and Technology Beijing under Grant GJJ2022-16 and JG2022M36 and the National Natural Science Foundation of China under Grant 61703035 and 61977004.

ORCID iDs

Xiaohe Li

Guoguang Wen

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

Altafini

(2012) Consensus problems on networks with antagonistic interactions. IEEE Transactions on Automatic Control 58(4): 935–946.

Altafini

Lini

(2014) Predictable dynamics of opinion forming for networks with antagonistic interactions. IEEE Transactions on Automatic Control 60(2): 342–357.

Bai

Wen

Rahmani

, et al. (2018) Consensus for the fractional-order double-integrator multi-agent systems based on the sliding mode estimator. IET Control Theory and Applications 12(5): 621–628.

Chai

Wang

Diao

, et al. (2022) Sampled-data-based dynamic event-triggered formation control for nonlinear multi-agent systems. Transactions of the Institute of Measurement and Control 44(14): 2719–2728.

Easley

Kleinberg

(2010) Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge: Cambridge University Press.

Girard

(2014) Dynamic triggering mechanisms for event-triggered control. IEEE Transactions on Automatic Control 60(7): 1992–1997.

Gong

(2020) Exponential bipartite consensus of fractional-order non-linear multi-agent systems in switching directed signed networks. IET Control Theory and Applications 14(17): 2582–2591.

Huang

Liu

Shan

(2021) Containment control of first-order multi-agent systems under pi coordination protocol. Algorithms 14(7): 209.

Jiang

Wen

Peng

, et al. (2022) Fully distributed pull-based event-triggered bipartite fixed-time output control of heterogeneous systems with an active leader. IEEE Transactions on Cybernetics 53: 3089–3100.

10.

Jin

Liu

, et al. (2023) Event-triggered distributed sliding mode control of fractional-order nonlinear multi-agent systems. Symmetry 15(6): 1247.

11.

Hua

You

, et al. (2021) Output feedback predefined-time bipartite consensus control for high-order nonlinear multiagent systems. IEEE Transactions on Circuits and Systems I: Regular Papers 68(7): 3069–3078.

12.

Liu

Zhao

, et al. (2023) Containment control for fractional-order multi-agent systems with mixed time delays. Mathematical Methods in the Applied Sciences 46(3): 3176–3186.

13.

Liu

Zhang

Wang

, et al. (2022) Adaptive containment control for fractional-order nonlinear multi-agent systems with time-varying parameters. IEEE/CAA Journal of Automatica Sinica 9(9): 1627–1638.

14.

Han

, et al. (2022) Adaptive bipartite output consensus for heterogeneous multi-agent systems via state/output feedback control. IEEE Transactions on Circuits and Systems II: Express Briefs 69(8): 3455–3459.

15.

Mahmoodi

Shojaei

(2022) Adaptive bipartite output consensus of nonlinear fractional-order multi-agent systems. International Journal of Systems Science 53(8): 1615–1638.

16.

Mishra

Upadhyay

Gupta

(2018) An approach to improve the performance of fractional-order sinusoidal oscillators. Chaos Solitons Fractals 116: 126–135.

17.

Podlubny

(1999) Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier.

18.

Ray

Sahoo

Das

(2016) Formulation and solutions of fractional continuously variable order mass–spring–damper systems controlled by viscoelastic and viscous–viscoelastic dampers. Advances in Mechanical Engineering 8(5): 168781401 6646505.

19.

Rong

Liu

Jiang

, et al. (2021) Observer-based multiagent bipartite consensus with deterministic disturbances and antagonistic interactions. IEEE Transactions on Cybernetics 52(11): 1–8.

20.

Sakthivel

Parivallal

Kong

, et al. (2023) Bipartite consensus for Takagi-Sugeno fuzzy uncertain multi-agent systems with gain fluctuations. IEEE Transactions on Signal and Information Processing Over Networks 9: 74–83.

21.

Shahvali

Azarbahram

Naghibi-Sistani

, et al. (2020) Bipartite consensus control for fractional-order nonlinear multi-agent systems: An output constraint approach. Neurocomputing 397: 212–223.

22.

Veeraian

Gandhi

Mangalanathan

(2018) Design and analysis of fractional order seismic transducer for displacement and acceleration measurements. Journal of Sound and Vibration 419: 123–139.

23.

Wang

Liu

Chen

(2022) Sampled-hold-based consensus control for second-order multiagent systems under aperiodically intermittent communication. IEEE Transactions on Circuits and Systems I: Regular Papers 69(9): 3794–3803.

24.

Wang

Dong

(2023) Reset event-triggered adaptive fuzzy consensus for nonlinear fractional-order multiagent systems with actuator faults. IEEE Transactions on Cybernetics 53(3): 1868–1879.

25.

Xia

Bai

, et al. (2023) Containment control for fractional order mass with nonlinearity and time delay via pull-based event-triggered mechanism. Applied Mathematics and Computation 454: 128094.

26.

Qin

(2023) Observer-based dynamic event-triggered bipartite consensus of discrete-time multi-agent systems. IEEE Transactions on Circuits and Systems II: Express Briefs 70(3): 1054–1058.

27.

Guan

, et al. (2022) Observer-based dynamic event-triggered semiglobal bipartite consensus of linear multi-agent systems with input saturation. IEEE Transactions on Cybernetics 53: 3139–3152.

28.

Yan

Zhang

Guo

, et al. (2022) Distributed adaptive time-varying formation control for Lipschitz nonlinear multi-agent systems. Transactions of the Institute of Measurement and Control 44(2): 272–285.

29.

Ahn

Shi

(2022) Event-triggered bipartite consensus for fuzzy multiagent systems under Markovian switching signed topology. IEEE Transactions on Fuzzy Systems 30(7): 2610–2620.

30.

Zhan

, et al. (2023) Fixed-time bipartite containment control of high-order multi-agent systems in the signed graph. IEEE Transactions on Circuits and Systems II: Express Briefs 70(4): 1520–1524.

31.

Zhang

Cai

Wang

, et al. (2020) Adaptive bipartite event-triggered output consensus of heterogeneous linear multiagent systems under fixed and switching topologies. IEEE Transactions on Neural Networks and Learning Systems 31(11): 4816–4830.

32.

Zhang

Luo

, et al. (2021a) Adaptive event-triggered leader-follower consensus of linear multiagent systems under directed graph with nonzero leader input. IEEE Transactions on Circuits and Systems II: Express Briefs 69(3): 1442–1446.

33.

Zhang

Wen

Peng

, et al. (2019) Group multiple lags consensus of fractional-order nonlinear leader-following multi-agent systems via adaptive control. Transactions of the Institute of Measurement and Control 41(5): 1313–1322.

34.

Zhang

Wen

Rahmani

, et al. (2021b) Cluster consensus of multi-agent systems with general linear and nonlinear dynamics via intermittent adaptive pinning control. Transactions of the Institute of Measurement and Control 43(6): 1337–1346.

35.

Zhao

Shan

Peng

, et al. (2023) Data-driven event-triggered bipartite consensus for multi-agent systems preventing dos attacks. IEEE Control Systems Letters 7: 1915–1920.